Henri Poincaré also saw the new geometries as revolutionary, but he disagreed with both Kant and Helmholtz. ![]() For Helmholtz, whether space is Euclidean or not is a question for experience.Īnd here's an even more radical view. So in contrast to Kant, Helmholtz thought that the postulates of geometry are not dictated by the human intellect or by logical necessity. It's hard to argue - as Helmholtz says, there is no geometrical experiment that you can do that will decide the question of which one of these worlds is the real one. The person in the mirror, your reflection, could equally insist that only the mirror world is real. But are you really that sure? The further-away people in the convex mirror appear smaller than they actually are, but if a yardstick were attached to them, the yardstick would also appear much smaller, so measurements within the mirror would be as consistent as they are in our world. You might say that the mirror is just an illusion, and that only our own world, the Euclidean world, can be real. Can we learn to order our perceptions in such a space? If you've made it safely so far using your car's convex mirror, the answer is "yes". Note the parallel lines that form the top and bottom edges of the supermarket shelves not always being the same distance apart. The mirror image is a three-dimensional non-Euclidean space. For example, look at the reflections in a convex mirror. ![]() It's hard to imagine ordering our perceptions in a non-Euclidean space, so perhaps non-Euclidean spaces are not really as real as Euclidean space.īut we can do it, said the physicist Hermann von Helmholtz. We order our perceptions in our mental space and its properties are the same for all human beings. Is it a thing? Is it a substance? Is it even real? The philosopher Immanuel Kant said that space is in the mind: when we make a geometrical construction, it's not the shapes we draw on paper that are important, but the way we see them in our mental space. Once you start thinking about the nature of space, you soon arrive at the question of what space actually is. Now let's turn to the impact of this realisation on human thought. But as was shown by the mathematician Bernhard Riemann (among others) there are many more non-Euclidean spaces besides the hyperbolic paraboloid, including positively curved spaces and spaces of three or more dimensions (you can find out more here). Carl-Friedrich Gauss, one of the discoverers of this fact, never even got up the nerve to publish his work on this subject. It's this realisation - that space doesn't have to be as Euclid and our intuition suggest, but could be otherwise - that 19th century thinkers found so revolutionary. It turns out that the hyperbolic paraboloid forms a perfectly decent geometric space. What's more, the blue and yellow parallel lines are not everywhere the same distance apart, as you would expect from parallel lines on the plane. But notice that the red and yellow lines are both parallels of the blue line, yet they pass through the same point. The lines drawn on the shape are the "straight lines" of the paraboloid: they are paths of shortest distance between points. (See the first article for an explanation.) If a straight line that falls on two straight lines makes the interior angles on the same side add up to less than two right angles, then the two straight lines, if produced indefinitely, meet on that side.All right angles are equal to one another. ![]() A circle can be constructed with any point as its centre.A finite straight line can be extended as long as desired.A straight line can be drawn from any point to any other point.
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